# Existence of harmonic symplectic structure on symplectic Riemannian manifold

This post is an expanded version of this MSE post.

Assume that $$(M, \omega)$$ is a symplectic manifold which is equiped with a Riemannian metric.

Is there a symplectic structure $$\omega '$$ which is a harmonic $$2$$-form? Can one choose such a $$\omega'$$ such that it would be de Rham cohomolgue to the initial form $$\omega$$?

We consider the following particular case:

For a Riemannian manifold $$(M,g)$$, we equip the tangent bundle $$TM$$ with the Sasaki metric $$g_s$$ and the natural symplectic structure $$\omega$$ arising from the canonical structure on the cotangent bundle.

Under which conditions on the Riemannian manifold $$(M,g)$$, is the symplectic $$2$$-form $$\omega$$ a harmonic form on $$(TM,g_s)$$?

• This is always true on a Kahler manifold: see eg here (the comment on the first answer is also relevant). – Mike Miller Sep 21 '18 at 19:27
• @MikeMiller thanks for your great comments. – Ali Taghavi Sep 22 '18 at 12:06
• Concerning your first question, whether on a symplectic manifold with given metric $g$ there always exists a harmonic symplectic form, I believe the answer should be negative. Though I don't know how to construct an example and don't know a reference. However, if you read what is written in 0.2.B' of Gromov's famous article ihes.fr/~gromov/wp-content/uploads/2018/08/945.pdf , it looks like he doesn't rule out the possibility that for some metrics on $\mathbb CP^2$ the harmonic two-form (unique up to scale) will vanish somewhere. – Dmitri Panov Dec 5 '18 at 22:49